Let's look at several aspects of problem solving in the mathematics classroom.

Successful problem solvers are able to see connections between problems they have solved in the past, and new, yet mathematically similar, problems. Many students need assistance in seeing the mathematical connections between problems as well as problem-solving strategies. For example, consider how you would solve the following problem:

If 3/4 of the students in a school turn in a contest entry, and if each entry has a 1/6 chance of being given a prize, what is the chance that a student randomly selected from the student body will be a prize winner?

This problem can be visualized by using a tree diagram. Each branch on a diagram represents the probability of a particular event taking place (there are 24 possible outcomes). The first set of four branches represents the probability that a randomly selected student is either in the group of students that turned in a contest entry (three branches: 1/4, 1/4, and 1/4), or in the group of students that didn't (1/4). The second set of branches represents the probability of each individual student being selected (1/6). Multiplying the two will give you the probability that a selected student will be on one of the first set of branches AND on one of the second set of branches in that group: (1/4) • (1/6) = 1/24. Finally, to get the probability that a student will be selected from the group of all students who turned in a contest entry, you need to add the probabilities on those three groups of branches: (1/24) + (1/24) + (1/24) = 3/24 = 1/8.

It may not be obvious that the area model for multiplying fractions shares some features with problems that involve finding the probability of an event happening. For example, an area model can help show that, in this case, there's a 3/24 or 1/8 chance that a randomly selected student will be a prize winner. The area of the rectangle with the dimensions of 3/4 and 1/6 will be equal to the product of these two fractions. To construct such a rectangle, first draw two squares with sides equal to 1. Then shade 3/4 within one square, and 1/6 within the other one. Overlapping the two squares will produce the desired rectangle. The area of the rectangle is equal to 3/24, since there are 24 pieces in total and three of them are in the overlapping area (double shaded):

To simplify this fraction, you can rearrange the double-shaded pieces into a different rectangle. First, you'll need to further subdivide the square, and rearrange the rectangle as shown below. The area of the new rectangle is equal to 1/8:

Rich problems offer multiple paths of access to various students and can be solved through a variety of strategies. They provide opportunities to compare, contrast, and critique the effectiveness of various methods and to analyze connections between methods.

It is more important for the teacher than for elementary students to be aware of a list of possible strategies and accompanying problems. Rather than teaching strategies, students can be presented with problems that are more likely to elicit use of specific strategies. When the opportunity arises, attention can be drawn to details and to the benefits of the strategy. Gradually, a class can be asked to reflect on such questions as, "Do you think this seems like a problem where we would want to make an organized list, or one where we should try 'guess and check'?" Being able to decide which strategy is likely to be appropriate for a given problem is a very important aspect of being a successful problem solver.

To see a list of strategies that may be helpful to students, review the summary page in Part A.